Type I/II errors and power

5 The time, in minutes, spent by customers at a particular gym has the distribution \(\mathrm { N } ( \mu , 38.2 )\). In the past the value of \(\mu\) has been 42.4. Following the installation of some new equipment the management wishes to test whether the value of \(\mu\) has changed.

1. State what is meant by a Type I error in this context.
2. The mean time for a sample of 20 customers is found to be 45.6 minutes. Test at the \(2.5 \%\) significance level whether the value of \(\mu\) has changed.

2 In the past, the time, in hours, for a particular train journey has had mean 1.40 and standard deviation 0.12 . Following the introduction of some new signals, it is required to test whether the mean journey time has decreased.

1. State what is meant by a Type II error in this context.
2. The mean time for a random sample of 50 journeys is found to be 1.36 hours. Assuming that the standard deviation of journey times is still 0.12 hours, test at the \(2.5 \%\) significance level whether the population mean journey time has decreased.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (b).

6 The masses of cereal boxes filled by a certain machine have mean 510 grams. An adjustment is made to the machine and an inspector wishes to test whether the mean mass of cereal boxes filled by the machine has decreased. After the adjustment is made, he chooses a random sample of 120 cereal boxes. The mean mass of these boxes is found to be 508 grams. Assume that the standard deviation of the masses is 10 grams.

1. Test at the \(2.5 \%\) significance level whether the mean mass of cereal boxes filled by the machine has decreased.
   Later the inspector carries out a similar test at the \(2.5 \%\) significance level, using the same hypotheses and another 120 randomly chosen cereal boxes.
   [0pt]
2. Given that the mean mass is now actually 506 grams, find the probability of a Type II error. [5]

4 In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.5 and standard deviation 5.2. A new challenge is devised and it is expected that students will take, on average, less than 25.5 minutes to complete this challenge. A random sample of 40 students is chosen and their mean time for the new challenge is found to be 23.7 minutes.

1. Assuming that the standard deviation of the time for the new challenge is 5.2 minutes, test at the \(1 \%\) significance level whether the population mean time for the new challenge is less than 25.5 minutes.
2. State, with a reason, whether it is possible that a Type I error was made in the test in part (a).

7 A researcher is investigating the actual lengths of time that patients spend with the doctor at their appointments. He plans to choose a sample of 12 appointments on a particular day.

1. Which of the following methods is preferable, and why?
   • Choose the first 12 appointments of the day.
   • Choose 12 appointments evenly spaced throughout the day.
   Appointments are scheduled to last 10 minutes. The actual lengths of time, in minutes, that patients spend with the doctor may be assumed to have a normal distribution with mean \(\mu\) and standard deviation 3.4. The researcher suspects that the actual time spent is more than 10 minutes on average. To test this suspicion, he recorded the actual times spent for a random sample of 12 appointments and carried out a hypothesis test at the 1\% significance level.
2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
3. Given that the total length of time spent for the 12 appointments was 147 minutes, carry out the test.
4. Give a reason why the Central Limit theorem was not needed in part (iii).

5 The time taken for a particular train journey is normally distributed. In the past, the time had mean 2.4 hours and standard deviation 0.3 hours. A new timetable is introduced and on 30 randomly chosen occasions the time for this journey is measured. The mean time for these 30 occasions is found to be 2.3 hours.

1. Stating any assumption(s), test, at the \(5 \%\) significance level, whether the mean time for this journey has changed.
2. A similar test at the \(5 \%\) significance level was carried out using the times from another randomly chosen 30 occasions.
   (a) State the probability of a Type I error.
   (b) State what is meant by a Type II error in this context.

6 A survey taken last year showed that the mean number of computers per household in Branley was 1.66 . This year a random sample of 50 households in Branley answered a questionnaire with the following results.

1. Calculate unbiased estimates for the population mean and variance of the number of computers per household in Branley this year.
2. Test at the \(5 \%\) significance level whether the mean number of computers per household has changed since last year.
3. Explain whether it is possible that a Type I error may have been made in the test in part (ii).
4. State what is meant by a Type II error in the context of the test in part (ii), and give the set of values of the test statistic that could lead to a Type II error being made.

7 The masses, in grams, of apples from a certain farm have mean \(\mu\) and standard deviation 5.2. The farmer says that the value of \(\mu\) is 64.6. A quality control inspector claims that the value of \(\mu\) is actually less than 64.6. In order to test his claim he chooses a random sample of 100 apples from the farm.

1. The mean mass of the 100 apples is found to be 63.5 g . Carry out the test at the \(2.5 \%\) significance level.
2. Later another test of the same hypotheses at the \(2.5 \%\) significance level, with another random sample of 100 apples from the same farm, is carried out. Given that the value of \(\mu\) is in fact 62.7 , calculate the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 In the past Laxmi's time, in minutes, for her journey to college had mean 32.5 and standard deviation 3.1. After a change in her route, Laxmi wishes to test whether the mean time has decreased. She notes her journey times for a random sample of 50 journeys and she finds that the sample mean is 31.8 minutes. You should assume that the standard deviation is unchanged.

1. Carry out a hypothesis test, at the \(8 \%\) significance level, of whether Laxmi's mean journey time has decreased.
   Later Laxmi carries out a similar test with the same hypotheses, at the \(8 \%\) significance level, using another random sample of size 50 .
2. Given that the population mean is now 31.5, find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A biologist wishes to test whether the mean concentration \(\mu\), in suitable units, of a certain pollutant in a river is below the permitted level of 0.5 . She measures the concentration, \(x\), of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. $$n = 50 \quad \Sigma x = 23.0 \quad \Sigma x ^ { 2 } = 13.02$$

1. Carry out a test at the \(5 \%\) significance level of the null hypothesis \(\mu = 0.5\) against the alternative hypothesis \(\mu < 0.5\).
   Later, a similar test is carried out at the \(5 \%\) significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.
2. Given that, in fact, the value of \(\mu\) is 0.4 , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The heights of one-year-old trees of a certain variety are known to have mean 2.3 m . A scientist believes that, on average, trees of this age and variety in her region are slightly taller than in other places. She plans to carry out a hypothesis test, at the \(2 \%\) significance level, in order to test her belief.

1. State the probability that she will make a Type I error.
   She takes a random sample of 100 such trees in her region and measures their heights, \(h \mathrm {~m}\). Her results are summarised below. $$n = 100 \quad \sum h = 238 \quad \sum h ^ { 2 } = 580$$
2. Carry out the test at the \(2 \%\) significance level. \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-10_2717_35_109_2012}
3. The scientist carries out the test correctly, but another scientist claims that she has made a Type II error. Comment on this claim.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

4 A study of a large sample of books by a particular author shows that the number of words per sentence can be modelled by a normal distribution with mean 21.2 and standard deviation 7.3. A researcher claims to have discovered a previously unknown book by this author. The mean length of 90 sentences chosen at random in this book is found to be 19.4 words.

1. Assuming the population standard deviation of sentence lengths in this book is also 7.3, test at the \(5 \%\) level of significance whether the mean sentence length is the same as the author's. State your null and alternative hypotheses.
2. State in words relating to the context of the test what is meant by a Type I error and state the probability of a Type I error in the test in part (i).

7 The number of cars caught speeding on a certain length of motorway is 7.2 per day, on average. Speed cameras are introduced and the results shown in the following table are those from a random selection of 40 days after this.

1. Calculate unbiased estimates of the population mean and variance of the number of cars per day caught speeding after the speed cameras were introduced.
2. Taking the null hypothesis \(\mathrm { H } _ { 0 }\) to be \(\mu = 7.2\), test at the \(5 \%\) level whether there is evidence that the introduction of speed cameras has resulted in a reduction in the number of cars caught speeding.
3. State what is meant by a Type I error in words relating to the context of the test in part (ii). Without further calculation, illustrate on a suitable diagram the region representing the probability of this Type I error.

4 People who diet can expect to lose an average of 3 kg in a month. In a book, the authors claim that people who follow a new diet will lose an average of more than 3 kg in a month. The weight losses of the 180 people in a random sample who had followed the new diet for a month were noted. The mean was 3.3 kg and the standard deviation was 2.8 kg .

1. Test the authors' claim at the \(5 \%\) significance level, stating your null and alternative hypotheses.
2. State what is meant by a Type II error in words relating to the context of the test in part (i).

7 The weights, \(X\) kilograms, of bags of carrots are normally distributed. The mean of \(X\) is \(\mu\). An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$

1. Carry out the test, at the \(10 \%\) significance level.
2. You may now assume that the population variance of \(X\) is 1.85 . The inspector weighs another random sample of 200 bags and carries out the same test at the \(10 \%\) significance level.
   (a) State the meaning of a Type II error in this context.
   (b) Given that \(\mu = 2.12\), show that the probability of a Type II error is 0.652 , correct to 3 significant figures.

6 Last year Samir found that the time for his journey to work had mean 45.7 minutes and standard deviation 3.2 minutes. Samir wishes to test whether his journey times have increased this year. He notes the times, in minutes, for a random sample of 8 journeys this year with the following results. $$\begin{array} { l l l l l l l l } 46.2 & 41.7 & 49.2 & 47.1 & 47.2 & 48.4 & 53.7 & 45.5 \end{array}$$ It may be assumed that the population of this year's journey times is normally distributed with standard deviation 3.2 minutes.

1. State, with a reason, whether Samir should use a one-tail or a two-tail test.
2. Show that there is no evidence at the \(5 \%\) significance level that Samir's mean journey time has increased.
3. State, with a reason, which one of the errors, Type I or Type II, might have been made in carrying out the test in part (ii).

4 In the past, the time taken by vehicles to drive along a particular stretch of road has had mean 12.4 minutes and standard deviation 2.1 minutes. Some new signs are installed and it is expected that the mean time will increase. In order to test whether this is the case, the mean time for a random sample of 50 vehicles is found. You may assume that the standard deviation is unchanged.

1. The mean time for the sample of 50 vehicles is found to be 12.9 minutes. Test at the \(2.5 \%\) significance level whether the population mean time has increased.
2. State what is meant by a Type II error in this context.
3. State what extra piece of information would be needed in order to find the probability of a Type II error.

4 In the past, the flight time, in hours, for a particular flight has had mean 6.20 and standard deviation 0.80 . Some new regulations are introduced. In order to test whether these new regulations have had any effect upon flight times, the mean flight time for a random sample of 40 of these flights is found.

1. State what is meant by a Type I error in this context.
2. The mean time for the sample of 40 flights is found to be 5.98 hours. Assuming that the standard deviation of flight times is still 0.80 hours, test at the \(5 \%\) significance level whether the population mean flight time has changed.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in your answer to part (ii).

5 The manufacturer of a certain type of biscuit claims that \(10 \%\) of packets include a free offer printed on the packet. Jyothi suspects that the true proportion is less than \(10 \%\). He plans to test the claim by looking at 40 randomly selected packets and, if the number which include the offer is less than 2 , he will reject the manufacturer's claim.

1. State suitable hypotheses for the test.
2. Find the probability of a Type I error.
   On another occasion Jyothi looks at 80 randomly selected packets and finds that exactly 6 include the free offer.
3. Calculate an approximate \(90 \%\) confidence interval for the proportion of packets that include the offer.
4. Use your confidence interval to comment on the manufacturer's claim. \(6 X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

3 In the past, Arvinder has found that the mean time for his journey to work is 35.2 minutes. He tries a different route to work, hoping that this will reduce his journey time. Arvinder decides to take a random sample of 25 journeys using the new route. If the sample mean is less than 34.7 minutes he will conclude that the new route is quicker. Assume that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.

1. Find the probability that a Type I error occurs.
2. Arvinder finds that the sample mean is 34.5 minutes. Explain briefly why it is impossible for him to make a Type II error.

7 The heights, in centimetres, of adult females in Litania have mean \(\mu\) and standard deviation \(\sigma\). It is known that in 2004 the values of \(\mu\) and \(\sigma\) were 163.21 and 6.95 respectively. The government claims that the value of \(\mu\) this year is greater than it was in 2004. In order to test this claim a researcher plans to carry out a hypothesis test at the \(1 \%\) significance level. He records the heights of a random sample of 300 adult females in Litania this year and finds the value of the sample mean.

1. State the probability of a Type I error. \includegraphics[max width=\textwidth, alt={}]{ff3433b0-baab-45e3-845e-56a794739bba-12_74_1577_557_322} ........................................................................................................................................ You should assume that the value of \(\sigma\) after 2004 remains at 6.95 .
2. Given that the value of \(\mu\) this year is actually 164.91 , find the probability of a Type II error.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

8 In order to test the effect of a drug, a researcher monitors the concentration, \(X\), of a certain protein in the blood stream of patients. For patients who are not taking the drug the mean value of \(X\) is 0.185 . A random sample of 150 patients taking the drug was selected and the values of \(X\) were found. The results are summarised below. $$n = 150 \quad \Sigma x = 27.0 \quad \Sigma x ^ { 2 } = 5.01$$ The researcher wishes to test at the \(1 \%\) significance level whether the mean concentration of the protein in the blood stream of patients taking the drug is less than 0.185 .

1. Carry out the test.
2. Given that, in fact, the mean concentration for patients taking the drug is 0.175 , find the probability of a Type II error occurring in the test.

7 A mill owner claims that the mean mass of sacks of flour produced at his mill is 51 kg . A quality control officer suspects that the mean mass is actually less than 51 kg . In order to test the owner's claim she finds the mass, \(x \mathrm {~kg}\), of each of a random sample of 150 sacks and her results are summarised as follows. $$n = 150 \quad \Sigma x = 7480 \quad \Sigma x ^ { 2 } = 380000$$

1. Carry out the test at the \(2.5 \%\) significance level.
   You may now assume that the population standard deviation of the masses of sacks of flour is 6.856 kg . The quality control officer weighs another random sample of 150 sacks and carries out another test at the 2.5\% significance level.
2. Given that the population mean mass is 49 kg , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The numbers of basketball courts in a random sample of 70 schools in South Mowland are summarised in the table.

1. Calculate unbiased estimates for the population mean and variance of the number of basketball courts per school in South Mowland.
   The mean number of basketball courts per school in North Mowland is 1.9 .
2. Test at the \(5 \%\) significance level whether the mean number of basketball courts per school in South Mowland is less than the mean for North Mowland.
3. State, with a reason, which of the errors, Type I or Type II, might have been made in the test in part (ii).

7 Bob is a self-employed builder. In the past his weekly income had mean \(\\) 546\( and standard deviation \)\\( 120\). Following a change in Bob's working pattern, his mean weekly income for 40 randomly chosen weeks was \(\\) 581\(. You should assume that the standard deviation remains unchanged at \)\\( 120\).

1. Test at the \(2.5 \%\) significance level whether Bob's mean weekly income has increased.
   Bob finds his mean weekly income for another random sample of 40 weeks and carries out a similar test at the \(2.5 \%\) significance level.
2. Given that Bob's mean weekly income is now in fact \(\\) 595$, find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.